LAS VEGAS (CBS News) -- When President Obama hits the stage at Del Sol High School in Las Vegas Tuesday to talk about immigration reform, he'll be employing a tactic that he's made standard practice: Rallying public support for a challenging policy agenda.

This time, however, the situation is a bit different: Unlike issues such as health care reform or tackling the deficit, the subject of immigration reform has lately inspired a significant level of cooperation and agreement. A group of eight senators yesterday unveiled a blueprint for comprehensive immigration reform legislation that they called "a major breakthrough."

While the Senate seemingly one-upped the president, releasing their plan before his event today, its proposal sits on shaky ground. There are still a number of Republicans opposed to the idea of "amnesty" for illegal immigrants, and even those who want to create a pathway to citizenship for undocumented residents could very easily get bogged down in the details. If the Senate's plan falls apart, Mr. Obama will be ready to step in with his own legislative outline, administration officials tell CBS News.

Mr. Obama Tuesday will lay out his vision for immigration reform - a vision he laid out in a major immigration speech in El Paso, Texas in May 2011. The blueprint introduced by the "group of eight" in the Senate is, in fact, very similar to the president's plan. Most critically, both lay out a pathway to citizenship for undocumented immigrants.

The president plans to make the case today that the Senate should act quickly, and he plans to address the Latino community specifically, according to administration officials. The latest CBS News poll shows that voters are increasingly on his side: 51 percent of Americans think undocumented immigrants working in the U.S. should be able to stay and apply for citizenship, up from 14 points since September 2011. Another 20 percent of Americans now say undocumented immigrants should stay as guest workers.

Monday, the White House credited the growing support for a pathway to citizenship -- among the public and in the Senate -- to Mr. Obama's efforts. Work on immigration reform is happening now, White House spokesman Jay Carney said, "because a consensus is developing in the country, a bipartisan consensus. And it's happening because the president has demonstrated significant leadership on this issue when the effort to achieve comprehensive immigration reform did not succeed in 2010, this president continued to press for it."

Tuesday in Las Vegas, Mr. Obama will have some of his strongest political allies backing his message. Close to 20 labor union leaders, along with working families and community partners, are holding their own even to press for reforms. "This moment calls for this president to step out boldly on this issue," NAACP President Ben Jealous, who is working with labor unions on immigration, told reporters yesterday.

Given the strained relations between the White House and Congress, and between Democrats and Republicans, some lawmakers' first instinct may be one of distrust any deals that materializing, especially if the president tries to stake out ownership of it.

"We are not going to just rubber stamp what the president of the United States has just decided," Sen. Jeff Sessions, R-Ala., said on the Senate floor yesterday. "We do need to ask some serious questions about any proposal, and maybe we can work forward with some legislation that would serve the national interest, maybe we can do it on a bipartisan basis, but it's going to take real attention to details, the details is what makes the difference and that is what I'm concerned about."

Senate Minority Leader Mitch McConnell, R-Ky., similarly said in a statement, "This effort is too important to be written in a back room and sent to the floor with a take-it-or-leave it approach. It needs to be done on a bipartisan basis and include ideas from both sides of the aisle." McConnell said he hoped the president today would endorse a "bipartisan approach rather than delivering another divisive partisan speech."

Even Sen. Marco Rubio, R-Fla., one of the senators in the "gang of eight" that's building the bipartisan blueprint, said last week on the Mark Levin radio show that "there's no such thing as an honest policy disagreement with the left anymore. If you disagree with them, you're a bad person." Still, Rubio and the rest of the "gang of eight" said they're more optimistic than ever that a comprehensive deal will be reached.

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